---
title: "Interior and boundary points in R²"
description: "Classify interior and boundary points in R² using strict inequalities, closed curves, neighborhoods, and edge cases."
canonical_url: "https://fanout.sh/ml-math/lessons/sets-interior-and-boundary-points-t-1qzjuqn"
md_url: "https://fanout.sh/ml-math/lessons/sets-interior-and-boundary-points-t-1qzjuqn.md"
access: "public"
---

# Interior and boundary points in R²

Classify interior and boundary points in R² using strict inequalities, closed curves, neighborhoods, and edge cases.

## Public overview

Classify real-line and plane points as interior, boundary, or neither by testing whether every small neighborhood stays inside, crosses both sides, or misses the set.

Neighborhoods as Open Intervals

A neighborhood makes the informal idea of points close to \(x 0\in\mathbb R\) precise. Choose a radius \(\delta 0\); the neighborhood consists of points whose distance from \(x 0\) is smaller than \(\delta\).

On the real line, that neighborhood is the open interval \[\left(x 0-\delta,x 0+\delta\right).\] Equivalently, it is the set \[\{x: x-x 0 <\delta\}.\]

For example, if \(x 0=2\) and \(\delta=0.1\), the neighborhood is \[(1.9,2.1).\] It contains points between \(1.9\) and \(2.1\), but it excludes both endpoints.

The endpoints are excluded because their distance from \(2\) is exactly \(0.1\), while the definition requires distance strictly smaller than \(\delta\). This strict inequality is why the neighborhood is called open.

Interior Points and the Interior of a Set

For a set of real numbers \(S\), a point \(x 0\) is an interior point of \(S\) if some neighborhood of \(x 0\) is fully contained in \(S\).

The word some is important: one sufficiently small neighborhood inside \(S\) proves that \(x 0\) is interior. Conversely, if every neighborhood around \(x 0\) includes at least one point outside \(S\), then \(x 0\) is not an interior point.

For \(S=[0,1)\), any point strictly between \(0\) and \(1\) is interior because a small enough open interval around that point stays inside \(S\). A point outside \(S\) cannot be interior because no neighborhood around it lies fully inside \(S\).

The endpoints are not interior points. Any neighborhood around \(0\) contains negative points outside \(S\), and any neighborhood around \(1\) contains points greater than or equal to \(1\), which are outside \([0,1)\).

The set of all interior points of \(S\) is called the interior of \(S\), written \(\operatorname{int}S\). For this example, \[\operatorname{int}([0,1))=(0,1).\]

Boundary Points Touch Both Sides of a Set

A point \(x 0\) is a boundary point of \(S\) if every neighborhood of \(x 0\) contains both points in \(S\) and points not in \(S\).

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